Wave with oscillating frequency

[Nikratio]

Hello,

In order to test-drive a data analysis program, I am looking for a function that generates sine waves with slowly oscillating frequency, i.e. the distance between the maxima should be slowly changing.

I thought that I could simply achieve this by using a function of the form
$$f(t) = \sin\Bigl( 2 \pi t \bigl[ 5 + \cos( 2 \pi t / 10 ) \bigr] \Bigl)$$
and expected to get something that oscillates with a frequency that slowly changes between 4 and 6 over when t goes from 0 to 10.

However, the function behaves in a very different way. It reaches a maximum of f(t)~0.5 at t~3 and has to minima very close to each other at t~4.2.

Can someone help me to understand (a) why the above function doesn't behave the way I expect it to and (b) what function I could use to generate the desired data?


Nikolaus

 

[uart]

Hi Nik, you've fallen for a really common mistake with regard to frequency modulation.

In general the instantaneous radian frequency (w=2 pi f) is the time rate of change of phase (phi), this of course is not the same thing as "phase" divided by "t" except for the special case where the phase is a linear function of "t", that is where [itex]\phi = 2 \pi f t[/tex].

So what you really want is,

$$\frac{d\phi}{d t} = 2 \pi ( 5 + \cos(\pi t /5) )$$

Which upon integration gives,

$$\phi = 10 \pi t + 10 \sin(\pi t /5)$$

Try using f(t)=sin(phi), with phi as given above and I think you'll find it does what you're looking for.

 

[Nikratio]

Yes, that was it! Thank you very much!